Multicomponent conjugation problems and auxiliary abstract boundary-value problems

V. I. Voititskiy ; N. D. Kopachevskiy ; P. A. Starkov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 34 (2009), pp. 5-44.

Bibliographie

[1] Agranovich M. S., Menniken R., “Spektralnye zadachi dlya uravneniya Gelmgoltsa so spektralnym parametrom v granichnykh usloviyakh na negladkoi poverkhnosti”, Matematich. sbornik, 190:1 (1999), 29–68 | MR | Zbl

[2] Agranovich M. S., Katsenelenbaum B. Z., Sivov A. N., Voitovich N. N., Generalized Method of Eigenoscillations in Difraction Theory, Wiley–VCH, Berlin, 1999, 380 pp. | MR | Zbl

[3] Agranovich M. S., Amosov B. A., Levitin M., “Spektralnye zadachi dlya sistemy Lame v gladkikh i negladkikh oblastyakh so spektralnym parametrom v kraevom uslovii”, Rossiiskii zhurnal matem. fiz., 6:3 (1999), 247–281 | MR | Zbl

[4] Starkov P. A., “Operatornyi podkhod k zadacham sopryazheniya”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 15(54):1 (2002), 58–62

[5] Starkov P. A., “Sluchai obschego polozheniya dlya operatornogo puchka, voznikayuschego pri issledovanii zadach sopryazheniya”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 15(54):2 (2002), 82–88

[6] Starkov P. A., “Issledovanie zadach sopryazheniya pri isklyuchitelnykh znacheniyakh fiksirovannogo parametra”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 16(55):2 (2003), 81–89

[7] Starkov P. A., “O bazisnosti sistemy sobstvennykh elementov v zadachakh sopryazheniya”, Tavricheskii vestnik informatiki i matematiki, 2003, no. 1, 118–131 | Zbl

[8] Kopachevskii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike. Evolyutsionnye i spektralnye zadachi, Nauka, M., 1989, 416 pp. | MR

[9] Kopachevsky N. D., Krein S. G., Operator Approach to Linear Problems of Hydrodynamics. Vol. 1: Self–adjoint Problems for an Ideal Fluid, Operator Theory: Advances and Applications, 128, Birkhäuser Verlag, Basel–Boston–Berlin, 2001, 384 pp. | MR | Zbl

[10] Kopachevsky N. D., Krein S. G., Operator Approach to Linear Problems of Hydrodynamics. Vol. 2: Nonselfadjoint Problems for Viscous Fluid, Operator Theory: Advances and Applications, 146, Birkhäuser Verlag, Basel–Boston–Berlin, 2003, 444 pp. | MR | Zbl

[11] Kopachevskii N. D., “Ob abstraktnoi formule Grina dlya troiki gilbertovykh prostranstv i ee prilozheniyakh k zadache Stoksa”, Tavricheskii vestnik informatiki i matematiki, 2004, no. 2, 52–80

[12] Kopachevskii N. D., Krein S. G., “Abstraktnaya formula Grina dlya troiki gilbertovykh prostranstv, abstraktnye kraevye i spektralnye zadachi”, Ukrainskii matem. vestnik, 1:1 (2004), 69–97 | MR | Zbl

[13] Gagliardo E., “Caratterizzazioni delle trace sulla frontiera relative ad alaine classi di funzioni in $n$ variabili”, Rendiconti del Seminare Matematico della Universita di Padova, 27 (1957), 284–305 | MR | Zbl

[14] Oben Zh.-P., Priblizhennoe reshenie ellipticheskikh kraevykh zadach, Mir, M., 1977, 384 pp. | MR

[15] Showalter R., Hilbert space methods for partial differential equations, Electronic Monographs in Differential Equations, 1994, 214 pp. | MR

[16] Lyantse V. E., Storozh O. G., Metody teorii neogranichennykh operatorov, Naukova dumka, Kiev, 1983, 212 pp. | MR

[17] Kopachevskii N. D., “Abstraktnaya formula Grina dlya smeshannykh kraevykh zadach”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 20(59):2 (2007), 3–12

[18] Agranovich M. S., “Remarks on Potential Spaces and Besov Spaces in a Lipschitz Domain and on Whitney Arrays on Its Bpundary”, Russian Journal of Math. Physics, 15:2 (2008), 146–155 | DOI | MR | Zbl

[19] Rychkov V. S., “On Restrictions and Extensions of the Besov and Triebel–Lizorkin Saces with Respect to Lipschitz Domains”, J. London Math. Soc., 60 (1999), 237–257 | DOI | MR | Zbl

[20] Paltsev B. V., “O smeshannoi zadache s neodnorodnymi granichnymi usloviyami dlya ellipticheskikh s parametrom uravnenii vtorogo poryadka v lipshitsevykh oblastyakh”, Matematicheskii sbornik, 187:4 (1996), 59–116 | MR | Zbl

[21] Krein S. G. (red.), Funktsionalnyi analiz, Nauka, M., 1972, 544 pp. | MR | Zbl

[22] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971, 372 pp. | Zbl

[23] Volevich L. R., Gindikin S. G., Obobschennye funktsii i uravneniya v svertkakh, Nauka, M., 1994, 336 pp. | MR | Zbl

[24] Gorbachuk V. I., Gorbachuk M. L., Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Naukova dumka, Kiev, 1984, 284 pp. | MR | Zbl

[25] Starkov P. A., “Primery mnogokomponentnykh zadach sopryazheniya”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 18(57):1 (2005), 89–94

[26] Radkevich E. V., “Ob usloviyakh suschestvovaniya klassicheskogo resheniya modifitsirovannoi zadachi Stefana (zakon Gibbsa–Tomsona)”, Matem. sb., 183:2 (1992), 77–101 | MR | Zbl

[27] Bazalii B. V., Degtyarev S. P., “O zadache Stefana s kineticheskim i klassicheskim usloviem na svobodnoi granitse”, UMZh, 44:2 (1992), 155–166 | MR

[28] Ercolano J., Schechter M., “Spectral theory for operators generated by elliptic boundary problems with eigenvalue parameter in boundary conditions. I”, Comm. Pure Appl. Math., 18 (1965), 83–105 | DOI | MR | Zbl

[29] Barkovskii V. V., “Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, sootvetstvuyuschikh obschim ellipticheskim zadacham s sobstvennym znacheniem v granichnykh usloviyakh”, UMZh, 19:1 (1967), 9–24 | MR

[30] Feschenko S. F., Lukovskii I. A., Rabinovich B. I., Dokuchaev L. V., Metody opredeleniya prisoedinennykh mass zhidkosti v podvizhnykh oblastyakh, Naukova dumka, Kiev, 1969, 216–224

[31] Below J., François G., “Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition”, Bull. Belgian Math. Soc. Simon Stevin, 12:4 (2005), 505–519 | MR | Zbl

[32] Voititskii V. I., “Abstraktnaya spektralnaya zadacha Stefana”, Uchenye zapiski Tavricheskogo natsionalnogo universiteta im. V. I. Vernadskogo. Ser. Matematika, Mekhanika, Informatika i Kibernetika, 19(58):2 (2006), 20–28

[33] Voititskii V. I., “O spektralnykh zadachakh, porozhdennykh zadachei Stefana s usloviyami Gibbsa–Tomsona”, Nelineinye granichnye zadachi, 17 (2007), 31–49

[34] Kopachevsky N. D., Voytitsky V. I., “On the modified spectral Stefan problem and its abstract generalizations”, Modern Analysis and Applications, Operator Theory: Advances and Applications, 191, Birkhäuser Verlag, Basel (Switzerland), 2009, 373–386 | MR

[35] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965, 448 pp. | MR

[36] Krein S. G., “O kolebaniyakh vyazkoi zhidkosti v sosude”, DAN SSSR, 159:2 (1964), 262–265 | MR

[37] Krein S. G., Laptev G. I., “K zadache o dvizhenii vyazkoi zhidkosti v otkrytom sosude”, Funktsionalnyi analiz i ego prilozheniya, 2:1 (1968), 40–50 | MR | Zbl

[38] Askerov N. K., Krein S. G., Laptev G. I., “Zadacha o kolebaniyakh vyazkoi zhidkosti i svyazannye s nei operatornye uravneniya”, Funktsionalnyi analiz i ego prilozheniya, 2:2 (1968), 21–31 | MR | Zbl

[39] Markus A. S., Matsaev V. I., “O bazisnosti chasti sobstvennykh i prisoedinennykh vektorov samosopryazhennykh operatornykh puchkov”, Matem. sbornik, 133(175):3 (1987), 293–313 | MR | Zbl

[40] Markus A. S., Matsaev V. I., “Teoremy o sravnenii spektrov lineinykh operatorov i spektralnye asimptotiki”, Trudy Mosk. matem. ob-va, 45, 1982, 133–181 | MR | Zbl

[41] Markus A. S., Matsaev V. I., “Teorema o sravnenii spektrov i spektralnye asimptotiki puchka M. V. Keldysha”, Matem. sbornik, 123(165):3 (1984), 391–406 | MR | Zbl

[42] Markus A. S., Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtiintsa, Kishinev, 1986, 260 pp. | MR | Zbl

[43] Kopachevskii N. D., “O svoistvakh bazisnosti sistemy sobstvennykh i prisoedinennykh vektorov samosopryazhennogo operatornogo puchka $I-\lambda A-\lambda^{-1}B$”, Funktsionalnyi analiz i ego prilozheniya, 15:2 (1981), 77–78 | MR | Zbl

[44] Grinshtein V. A., Kopachevskii N. D., “O $p$-bazisnosti sistemy sobstvennykh elementov samosopryazhennoi operator-funktsii”, Tez. XV Vsesoyuzn. shkoly po teorii operatorov v funktsionaln. prostranstvakh, Chast I (Ulyanovsk, 5–12 sentyabrya 1990 g.), 12

[45] Azizov T. Ya., Kopachevsky N. D., “On basicity of the system of eigen and associated elements of the S. G. Krein's problem on normal oscillations of a viscous fluid in an open vessel”, Spektralnye i evolyutsionnye zadchi, Sborn., Tezisy lektsii i dokladov III Krymskoi osennei matematicheskoi shkoly-simpoziuma, Vyp. 3, Krym, Laspi, 1994, 38–39

[46] Kopachevskii N. D., “O $p$-bazisnosti sistemy kornevykh vektorov samosopryazhennogo operatornogo puchka $I-\lambda A-\lambda^{-1}B$”, Funktsionalnyi analiz i prikl. matemtika, Sborn. nauchn. trudov, Naukova Dumka, Kiev, 1982, 43–55 | MR

[47] Kopachevskii N. D., Nemirskaya O. I., O $p$-bazisnosti sistemy sobstvennykh elementov samosopryazhennoi operator-funktsii, Rus. Deponir. v UkrINTEI 16.12.92, No 1969 – Uk. 92, Stmferopolskii un-t, Simferopol, 1992, 10 pp.

[48] Grinshtein V. A., Kopachevskii N. D., K zadache o spektre okaimlennogo samosopryazhennogo operatora, Deponir. v UkrNIINTI 03.01.89, No 109 – Uk 89, 25 pp.

[49] Abramov Yu. Sh., Variatsionnye metody v teorii operatornykh puchkov. Spektralnaya optimizatsiya, Izd-vo LGU, L., 1983, 178 pp. | MR | Zbl

[50] Smirnov V. I., Kurs vysshei matematiki, Tom 5, GITTL, M., 1960, 656 pp. | MR

[51] Azizov T. Ya., Iokhvidov I. S., Osnovy teorii lineinykh operatorov v prostranstve s indefinitnoi metrikoi, Nauka, M., 1986, 352 pp. | MR

[52] Chueshov I., Introduction to the Theory of Infinite-Dimentional Dissipative Systems, Ada, Kharkov, 1999, 436 pp. (in Russian) | MR

[53] Chueshov I., Eller M., Lasiecka I., “Finite dimentionality of the attractor for a Semilinear Wave Equation with Nonlinear Boundary Dissipation”, Communications in PDE, 29:11–12 (2004), 1847–1876 | MR | Zbl

[54] Kopachevsky N. D., “On abstract boundary value problems with surface dissipation of an energy”, Analysis and PDE's (in honor of prof. Bogdan Bojarsky), Abstracts of the conference (Bedlevo, Poland, June 18–24, 2006), 23

[55] Andronova O. A., Kopachevskii N. D., “Nachalno-kraevye i spektralnye zadachi s poverkhnostnoi dissipatsiei energii”, Vseukrainsk. nauchn. konf. molodykh uchenykh i studentov po diff. uravn. i ikh prilozheniyam, posvyaschennoi 100–letnemu yubileyu Ya. B. Lopatinskogo, Tez. dokl. (Donetsk, 6–7 dekabrya 2006 g.), 12–13

[56] Andronova Olga, Kopachevsky Nikolay, “Operator approach to dynamic systems with surface disipation of an energy”, Intern. conf. “Modern Analysis and Applications” (MAA–2007), dedicated to the centenary of Mark Krein, Book of abstracts (Odessa, Ukraine, April 9–14, 2007), 11

[57] Kopachevsky N. D., Starkov P. A., “Operator approach to transmission problems for Helmholtz equation”, Ukr. Congr. of Mathematics, Nonlinear PDE, Book of abstracts (Kyiv, 22–28 August, 2001), 68

[58] Kopachevsky N. D., Starkov P. A., “Abstract Green's Formula and Transmission Problems”, Intern. Conf. on Functional Analysis and its Applications (dedicated to the 110th anniversary of Stefan Banach), Book of abstracts (Lviv, Ukraine, May 28–31, 2002), 112–113

[59] Kopachevsky N. D., Starkov P. A., Voytitsky V. I., “Abstract Green's Formula and Spectral Transmission Problems”, Intern. Conf. Nonlinear PDE, Book of abstracts (Yalta, Crimea, Ukraine, September 10–15, 2007), 40–41